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American Mathematical Monthly Contents—April 2011

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ARTICLES

Integral Apollonian Packings

Peter Sarnak
We review the construction of integral Apollonian circle packings. There are a number of Diophantine problems that arise in the context of such packings. We discuss some of them and describe some recent advances.

Solving Cubics With Creases: The Work of Beloch and Lill

Thomas C. Hull
Margharita P. Beloch was the first person, in 1936, to realize that origami (paperfolding) constructions can solve general cubic equations and thus are more powerful than straightedge and compass constructions. We present her proof. In doing this we use a delightful (and mostly forgotten?) geometric method due to Eduard Lill for finding the real roots of polynomial equations.

A Cubic Analogue of the Jacobsthal Identity

Heng Huat Chan, Ling Long, and YiFan Yang
It is well known that if p is a prime such that p≡ 1 (mod 4), then p can be expressed as a sum of two squares. Several proofs of this fact are known and one of them, due to E. Jacobsthal, involves the identity p=x2 + y2, with x and y expressed explicitly in terms of sums involving the Legendre symbol. These sums are now known as the Jacobsthal sums. In this short note, we prove that if p≡ 1 (mod 6), then 3p= u2 + uv = v2 for some integers u and v using an analogue of Jacobsthal’s identity.

Artifacts for Stamping Symmetric Designs

H. M. Hilden, J. M. Montesinos, D. M. Tejada, and M. M. Toro
It is well known that there are 17 crystallographic groups that determine the possible tessellations of the Euclidean plane. We approach them from an unusual point of view. Corresponding to each crystallographic group there is an orbifold. We show how to think of the orbifolds as artifacts that serve to create tessellations.

A New Singular Function

Jaume Paradís, Pelegrí Viader, and Lluís Bibiloni
A new continuous strictly increasing singular function is described with the help of the ternary and binary systems for real number representation; in this, our function is similar to Cantor’s function, but in other aspects it is quite unusual. We are able to determine a condition to identify many points for which the derivative vanishes or is infinite; for other singular functions constructed with the help of a system of representation of real numbers, this condition depends on some metrical properties of the growth of averages of the sum of all the digits of the representation, but in the case of this new function, it depends on the frequency of occurrence of the digit 2 in the usual ternary expansion of a number.

NOTES

A Remark on Euclid’s Theorem on the Infinitude of the Primes

Roger Cooke
We examine and update an 1889 application of the theory of finite abelian groups to prove that there are at least n- 1primes between the nth prime and the product of the first n primes.

When Is a Polynomial a Composition of Other Polynomials?

James Rickards
In this note we explore when a polynomial f(x) can be expressed as a composition of other polynomials. First, we give a necessary and sufficient condition on the roots of f(x). Through a clever use of symmetric functions we then show how to determine if f(x) is expressible as a composition of polynomials without needing to know any of the roots of f(x).

A Top Hat for Moser’s Four Mathemagical Rabbits

Pieter Moree
If the equation 1k = 2k + --- + (m-2)k = mk has a solution with k ≥ 2, then. Leo Moser showed this in 1953 by remarkably elementary methods. His proof rests on four identities he derives separately. It is shown here that Moser’s result can be derived from a von Staudt-Clausen type theorem (an easy proof of which is also presented here). In this approach the four identities can be derived uniformly. The mathematical arguments used in the proofs were already available during the lifetime of Lagrange (1736–1813).

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